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Difficulty: EasySet Theory Concepts and Venn Diagrams

A museum surveyed 150150 visitors regarding their attendance at two special exhibitions: a Fine Art exhibit and a Natural History exhibit. Among the visitors surveyed, 8585 attended the Fine Art exhibit, 7070 attended the Natural History exhibit, and 2020 attended neither exhibit. How many visitors attended both the Fine Art exhibit and the Natural History exhibit?

  1. A
    55
  2. 2525Answer
  3. C
    4545
  4. D
    6565
  5. E
    155155

Answer

The correct answer is 2525 visitors.
Using the inclusion-exclusion formula Total=∣A∣+∣B∣−∣A∩B∣+Neither\text{Total} = |A| + |B| - |A \cap B| + \text{Neither}, we substitute the given values: 150=85+70−∣A∩B∣+20150 = 85 + 70 - |A \cap B| + 20. Simplifying gives 150=175−∣A∩B∣150 = 175 - |A \cap B|, which yields ∣A∩B∣=25|A \cap B| = 25. Thus, 2525 visitors attended both exhibits.

Step-by-Step Solution

1
Identify the given set values and formula.
Total visitors =150= 150, Fine Art attendees ∣A∣=85|A| = 85, Natural History attendees ∣H∣=70|H| = 70, Neither =20= 20.
The principle of inclusion-exclusion for two sets states that Total=∣A∣+∣H∣−∣A∩H∣+Neither\text{Total} = |A| + |H| - |A \cap H| + \text{Neither}.
2
Calculate the number of visitors who attended at least one exhibit.
∣A∪H∣=150−20=130|A \cup H| = 150 - 20 = 130.
Subtracting those who attended neither exhibit from the total population yields the total number of unique visitors who attended at least one of the two exhibits.
3
Solve for the intersection ∣A∩H∣|A \cap H|.
130=85+70−∣A∩H∣  ⟹  130=155−∣A∩H∣  ⟹  ∣A∩H∣=25130 = 85 + 70 - |A \cap H| \implies 130 = 155 - |A \cap H| \implies |A \cap H| = 25.
Subtracting the union ∣A∪H∣|A \cup H| from the sum of the individual sets ∣A∣+∣H∣|A| + |H| eliminates the double-counted intersection.

Key Concept

Two-Set Principle of Inclusion-Exclusion
Estimated Time:1m 0s
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Topics

Set Theory Concepts and Venn Diagrams

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